On the mystery of the cosmic vacuum energy density

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After a short history of the Λ-term it is explained why the (effective) cosmological constant is expected to obtain contributions from short-distance physics, corresponding to an energy at least as large as the Fermi scale. The actual tiny value of the cosmological constant by particle physics standards represents, therefore, one of the deepest mysteries of present-day fundamental physics. Recent proposals of an approach to the cosmological constant problem which make use of (large) extra dimensions are briefly discussed. Cosmological models with a dynamical Λ, which attempt to avoid the disturbing cosmic coincidence problem, are also reviewed.